# Prove $1-\xi_p$ is irreducible in $\mathbb{Z}[\xi_p]$

I am trying to prove that $$1-\xi_p$$ is irreducible in $$\mathbb{Z}[\xi_p]$$, where $$\xi_p$$ is a primitive $$p$$-root of unity.

I am not sure how to do this. I have calculated the norm of $$1-\xi_p$$ and I know it is equal to the prime $$p$$. But I am not sure if I can conclude from this. I know that if an element of $$\mathbb{Z}[\xi_p]$$ is a unit then its norm is $$+1$$ or $$-1$$. But I am not sure whether having norm $$+1$$ or $$-1$$ implies that an element is a unit, which is what I would need to be able to conclude that $$1-\xi_p$$ is irreducible from the fact that it has norm a prime.

Any clarification will be very useful. Thanks in advance.

• "But I am not sure whether having norm $+1$ or $-1$ implies that an element is a unit," - yes, it is true. See this duplicate. Or this one. Dec 29, 2022 at 18:55

## 1 Answer

Yes: For a number field $$k$$, any $$x\in {\cal O}_k$$ with $$N_{k/\mathbb{Q}}(x) = \pm 1$$ is invertible. For if $$f(x) = x^n + \cdots + a_1 x + a_0$$ is the minimal polynomial of $$x$$, then $$x(x^{n-1} + \cdots + a_1) = -a_0$$ with $$-a_0 = \pm N_{k/\mathbb{Q}}(x)\in {\cal O}_k^\times$$.